<h2>Problem 183</h2>
<div style="color:#666;font-size:80%;">22 February 2008</div><br />
<div class="problem_content">
<p>Let N be a positive integer and let N be split into <var>k</var> equal parts, <var>r</var> = N/<var>k</var>, so that N = <var>r</var> + <var>r</var> + ... + <var>r</var>.<br />
Let P be the product of these parts, P = <var>r</var> <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> <var>r</var> <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> ... <img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' /> <var>r</var> = <var>r</var><img src="" style="display:none;" alt="^(" /><sup><var>k</var></sup><img src="" style="display:none;" alt=")" />.</p>

<p>For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.2<img src="" style="display:none;" alt="^(" /><sup>5</sup><img src="" style="display:none;" alt=")" /> = 51.53632.</p>

<p>Let M(N) = P<img src="" style="display:none;" alt="_(" /><sub>max</sub><img src="" style="display:none;" alt=")" /> for a given value of N.</p>

<p>It turns out that the maximum for N = 11 is found by splitting eleven into four equal parts which leads to P<img src="" style="display:none;" alt="_(" /><sub>max</sub><img src="" style="display:none;" alt=")" /> = (11/4)<img src="" style="display:none;" alt="^(" /><sup>4</sup><img src="" style="display:none;" alt=")" />; that is, M(11) = 14641/256 = 57.19140625, which is a terminating decimal.</p>

<p>However, for N = 8 the maximum is achieved by splitting it into three equal parts, so M(8) = 512/27, which is a non-terminating decimal.</p>

<p>Let D(N) = N if M(N) is a non-terminating decimal and D(N) = -N if M(N) is a terminating decimal.</p>

<p>For example, &Sigma;D(N) for 5 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> N <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 100 is 2438.</p>

<p>Find &Sigma;D(N) for 5 <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> N <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 10000.</p>

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